Graduate Assumptions of the Bell theorem

[vanhees71]

Which means that, while in principle yes, all the information is in that system, in practice most of that information is inaccessible to us. We certainly can't do quantum state tomography "from the outside" on a whole ensemble of identically prepared system + measuring apparatus + environment in order to find out exactly which pure state is being prepared.

Also, considering this closed system doesn't solve the measurement problem either, since this closed system should just undergo unitary evolution all the time, since it is not interacting with anything, and therefore we end up at something like the MWI. I see that this is more or less where you ended up in your exchange with @vanhees71.

In addition we cannot even prepare a macroscopic system in exactly the same microscopic (pure) state to begin with. All we can do and all we need to do is to prepare the macroscopic system with sufficient accuracy by determining its macroscopic (relevant) observables with sufficient accuracy.

 

[WernerQH]

With the theory you presented, can you explain why typical macro pointers don't distinguish a cat in the state $|dead\rangle+|alive\rangle$ from the cat in the state $|dead\rangle-|alive\rangle$?

I think this is a blatant misuse of notation. Had Schrödinger replaced the poor cat by a calorimeter, could you even conceive of a coherent superposition of two states |14°C> and |15°C> ? For a tiny drop of $1 {\rm mm}^3$ water one finds an entropy increase $S/k = {\rm ln} W$ of about $10^{18}$, and that's just the logarithm of the tremendous factor by which the 15°C states are more numerous than the 14°C states. One frequently sees these fictitious kets "correctly" normalized with a factor $1/\sqrt 2$, but this normalization looks strikingly suspicious. You have wisely refrained from adding this factor, but I still find it hard to see how the two states could have equal weight in a superposition. I think it's meaningless to represent such states by any kind of wave function.

 

[Kolmo]

I think it's meaningless to represent such states by any kind of wave function

As mentioned above by vanhees71 and as seen in many models of measurements, the states of the device are of course in fact high-entropy mixed states, usually constructed by maxent methods or similar. The temperature example you give being a clear case, as a state of some temperature $T$ is a Gibb's state and not a pure state. To say nothing of other macroscopic quantities entering into the definition of the macrostate.

 

[Fra]

No. Models of measurement equally model POVMs, Weak Measurements and so on. The Curie-Weiss model of measurement which is the default "very detailed" measurement model naturally produces POVMs. So it's not restricted to optimal measurements.

Mmmm... thanks.

Does this argument include also the defintion of "weak measurements" that can give large disturbances on the observed system on the measured system, but at low probability (so that it "on average" is still weak)? instead of what i think is more common, that ALL outcomes give small disturbance? and extremal variability is truncated.I suspect this may influence what is meany by compatibibility. I.e are incompatibilities at low probabilities allowed?

/Fredrik

 

[Demystifier]

I think this is a blatant misuse of notation. Had Schrödinger replaced the poor cat by a calorimeter, could you even conceive of a coherent superposition of two states |14°C> and |15°C> ? For a tiny drop of $1 {\rm mm}^3$ water one finds an entropy increase $S/k = {\rm ln} W$ of about $10^{18}$, and that's just the logarithm of the tremendous factor by which the 15°C states are more numerous than the 14°C states. One frequently sees these fictitious kets "correctly" normalized with a factor $1/\sqrt 2$, but this normalization looks strikingly suspicious. You have wisely refrained from adding this factor, but I still find it hard to see how the two states could have equal weight in a superposition. I think it's meaningless to represent such states by any kind of wave function.

Almost every sentence here contains a conceptual error, but let me concentrate on (what seems to me) the essence of your argument. You argue that if one state has much larger entropy than the other, then the probabilities of those two states cannot be the same. But that's wrong. It would be right in a statistical equilibrium (which maximizes entropy under given constraints), but in general we don't need to have a statistical equilibrium.

For example, I can prepare a spin-1/2 particle in an eigenstate of spin in the x-direction and then measure its spin in the z-direction. If the z-spin is up, I prepare the drop in the state |14°C>. If the z-spin is down, I prepare the drop in the state |15°C>. In this way, the states |14°C> and |15°C> have the same statistical weights.

 

[Kolmo]

Does this argument include also the defintion of "weak measurements"

Weak measurements are just POVMs whose Kraus operators are close to the identity and possibly followed by post-selection. So yes essentially.

 

[Kolmo]

I prepare the drop in the state |14°C>

It's important though that there is no such pure state, it is necessarily mixed. This is an important point in detailed models of measurement.

 

[Demystifier]

closed system ... is not interacting with anything,

It is interacting with itself.

 

[WernerQH]

For example, I can prepare a spin-1/2 particle in an eigenstate of spin in the x-direction and then measure its spin in the z-direction. If the z-spin is up, I prepare the drop in the state |14°C>. If the z-spin is down, I prepare the drop in the state |15°C>. In this way, the states |14°C> and |15°C> have the same statistical weights.

But then you are no longer talking about unitary evolution of kets, but ordinary classical statistical physics.

 

[Demystifier]

It's important though that there is no such pure state, it is necessarily mixed.

That's one of the reasons why I said that almost any sentence in his post contains a conceptual error. But there is a way to associate something like a temperature with a pure state, e.g. by studying how energy is distributed in this state.

 

[Demystifier]

But then you are no longer talking about unitary evolution of kets, but ordinary classical statistical physics.

But if the spin-1/2 particle entangled with the drop are isolated from the rest of environment, then we have a coherent superposition.

 

[Lord Jestocost]

But physicists can't agree on what constitutes a "measurement".

Correct!

Maximilian Schlosshauer/1/ clearly identifies the measurement problem in the following way:

“But what exactly is the measurement problem? I have found that everyone seems to have a somewhat different conception of the affair. One way of identifying the root of the problem is to point to the apparent dual nature and description of measurement in quantum mechanics. On the one hand, measurement and its effect enter as a fundamental notion through one of the axioms of the theory. On the other hand, there’s nothing explicitly written into these axioms that would prevent us from setting aside the axiomatic notion of measurement and instead proceeding conceptually as we would do in classical physics. That is, we may model measurement as a physical interaction between two systems called “object” and “apparatus” — only that now, in lieu of particles and Newtonian trajectories, we’d be using quantum states and unitary evolution and entanglement-inducing Hamiltonians.

What we would then intuitively expect — and perhaps even demand — is that when it’s all said and done, measurement-as-axiom and measurement-as-interaction should turn out to be equivalent, mutually compatible ways of getting to the same final result. But quantum mechanics does not seem to grant us such simple pleasures. Measurement-as-axiom tells us that the post-measurement quantum state of the system will be an eigenstate of the operator corresponding to the measured observable, and that the corresponding eigenvalue represents the outcome of the measurement. Measurement-as-interaction, by contrast, leads to an entangled quantum state for the composite system-plus-apparatus. The system has been sucked into a vortex of entanglement and no longer has its own quantum state. On top of that, the entangled state fails to indicate any particular measurement outcome.

So we’re not only presented with two apparently mutually inconsistent ways of describing measurement in quantum mechanics, but each species leaves its own bad taste in our mouth. When confronted with measurement-as-axiom, many people tend to wince and ask: “But ... what counts as a measurement? Why introduce a physical process axiomatically? What makes the quantum state collapse?” And so on. But measurement-as-interaction delivers no ready-made remedy either. As we have seen, the interaction leads to nothing that would resemble the outcome of a measurement in any conventional sense of the word.”
[bold by LJ]

/1/ M. Schlosshauer (ed.), Elegance and Enigma, The Quantum Interviews, Springer-Verlag Berlin Heidelberg 2011, pp. 141-142

 

[Kolmo]

That's one of the reasons why I said that almost any sentence in his post contains a conceptual error. But there is a way to associate something like a temperature with a pure state, e.g. by studying how energy is distributed in this state.

In general for a selection of properties one can construct pure states that give similar results to Gibb's states, but many statistical mechanical properties won't give the same results and in general the entanglement measures are not correct.
If one is only concentrating on a small selection of properties and not too interested in dynamics then indeed there are a few calculational methods using pure states.

This isn't to disagree with anything you said, just more an "out of interest" thing.

 

[Kolmo]

Measurement-as-axiom tells us that the post-measurement quantum state of the system will be an eigenstate of the operator corresponding to the measured observable, and that the corresponding eigenvalue represents the outcome of the measurement. Measurement-as-interaction, by contrast, leads to an entangled quantum state for the composite system-plus-apparatus.

I kind of understand what is being said here but two points.

(a) In general the state afterward is not some eigenvalue of a observable, but rather updated via a Kraus operator. This isn't too important since the text might be focusing purely on von Neumann style measurements.

(b) There's nothing exceptional here that isn't in any statistical theory. I can apply the exact same arguments to the evolution of an option price under Black-Scholes. There as well "observation" of the option price does not produce the exact same answer as evolution under the Black-Scholes equations where one includes the "price evaluator" in the model.
In fact you get the exact same "issues" in a statistical model of classical particle interactions. Or even of a dice roll if one "included the measuring device".

 

[WernerQH]

But if the spin-1/2 particle entangled with the drop are isolated from the rest of environment, then we have a coherent superposition.

The spin state of an electron and the thermal state of a drop are very different things. It is beyond me how you can give meaning to empty symbolism such as |dead> + |alive>.

 

[Demystifier]

It is beyond me how you can give meaning to empty symbolism such as |dead> + |alive>.

If it was not a cat but a virus, would it still be empty for you?

 

[Kolmo]

Now there are old arguments from Ludwig in:
G. Ludwig: “Die Grundlagen der Quantenmechanik”, Springer, Berlin 19541
that such Q in most cases probably can't be performed at all as the coupling Hamiltonians needed to enact them aren't physical at all

Addendum:
More explicit in:
G.Ludwig: “Geloeste und ungeloeste Probleme des Messprozesses in der Quantenmechanik”, in “W.Heisenberg und die Physik unserer Zeit”, ed. F.Bopp, Vieweg, Braunschweig 1961

 

[Fra]

What we would then intuitively expect — and perhaps even demand — is that when it’s all said and done, measurement-as-axiom and measurement-as-interaction should turn out to be equivalent, mutually compatible ways of getting to the same final result.

I agree this is more or less the the core problem.

Another thing I would by the same reasoning "inutitively expect and perhaps demand" is to reconstruct the hamiltonians of the standard model from the inference rules of the RIGHT inference/measurement theory, in the right context.

Anything less, and we will still be here in hundred years discussing the same thing.

/Fredrik

 

[PeterDonis]

It is interacting with itself.

No, it isn't; such a statement makes no sense. You might be able to partition the closed system into subsystems that interact with each other, but any such partitioning is basis dependent. But the system as a whole can't "interact with itself"; interaction requires at least two systems.

 

[PeterDonis]

But if the spin-1/2 particle entangled with the drop are isolated from the rest of environment

Which they can't be in practice. And the claim that they can be in principle, even if not in practice, is a claim that cannot be tested experimentally, so it should be viewed with great caution.

 

[Kolmo]

If it was not a cat but a virus, would it still be empty for you?

For a literal virus I wouldn't say it was "empty" as such, but certainly an incorrect state assignment, since an actual virus will be in thermal equilibrium with some environment, emitting EM radiation and so forth. Perhaps WernerQH means simply that. A state that is so wrong could be said to have little practical meaning.

 

[Lord Jestocost]

A state that is so wrong could be said to have little practical meaning.

Why is a superpositon state as |dead> + |alive> so "wrong". Only because we don’t know at present how to look for such a state doesn't justify statements like this, not at all from a scientific point of view.

 

[Kolmo]

Why is a superpositon state as |dead> + |alive> so "wrong". Only because we don’t know at present how to look for such a state doesn't justify statements like this, not at all from a scientific point of view.

As I said above, a virus in real life is embedded in a thermal environment, it's characterised by values for macroscopic observables and so forth. All of these things give a mixed state as the correct state, not a pure state.

 

[Demystifier]

But the system as a whole can't "interact with itself"; interaction requires at least two systems.

What about $\phi^4$ theory, or gravity, or Yang-Mills theory? Aren't those self-interacting theories of scalar field, metric-tensor field and gauge field, respectively?

 

[vanhees71]

But then you are no longer talking about unitary evolution of kets, but ordinary classical statistical physics.

For a closed system the states (no matter whether they are pure or mixed ones) and observable operators evolve by unitary time evolution. The choice how they do that is pretty arbitrary. That's known as the choice of the picture of time evolution. What's of course independent are all measurable quantities like probabilities for the outcome of measurements, expectation values/correlation functions, S-matrix elements in scattering theory, etc.

 

[vanhees71]

No, it isn't; such a statement makes no sense. You might be able to partition the closed system into subsystems that interact with each other, but any such partitioning is basis dependent. But the system as a whole can't "interact with itself"; interaction requires at least two systems.

Well, yes. Take a lonely hydrogen atom within non-relativistic QT. It consists of a proton and an electron interacting with each other via the Coulomb interaction. That's an example for what you usually call an interacting closed system.

 

[vanhees71]

What about $\phi^4$ theory, or gravity, or Yang-Mills theory? Aren't those self-interacting theories of scalar field, metric-tensor field and gauge field, respectively?

Of course. There's a clear meaning of what interacting closed systems are. I think this is again just some semantical discussion about words, whose meaning is clearly established in the scientific community.

 

[WernerQH]

For a closed system the states (no matter whether they are pure or mixed ones) and observable operators evolve by unitary time evolution. The choice how they do that is pretty arbitrary. That's known as the choice of the picture of time evolution. What's of course independent are all measurable quantities like probabilities for the outcome of measurements, expectation values/correlation functions, S-matrix elements in scattering theory, etc.

A closed system is an idealization. There's no such thing in the real world, let alone one containg a cat (or a virus).

So you can detect a modicum of sense in an expression like |dead> + i × |alive> ?

 

[Lord Jestocost]

As I said above, a virus in real life is embedded in a thermal environment, it's characterised by values for macroscopic observables and so forth. All of these things give a mixed state as the correct state, not a pure state.

What do you think is the effect when a superposition state interacts with its environment? The quantum mechanical formalism is here unambiguous: You get an entangled quantum state for the composite “system-plus-environment”. Maybe, the interaction between system and environment scrambles up the phases so that it would be impossible, from a practical point of view, to unscramble them. However, the superposition state does not evolve by the Schrödinger equation into a mixed one. With all due respect, this statement is wrong.

 

[vanhees71]

I think a state $|\text{alive} \rangle + |\text{alive} \rangle$ is simply a nonsensical expression, because $|\text{alive}/\text{dead} \rangle$ simply don't exist.